So... this bugs you. Okay. It's bizarre behavior where someone you don't like much is trying to gain benefit by association with you. I can see how that would bug you. So... what is the result you wish to achieve, and why?
Perhaps you just want her to stop doing that thing, without having to get into some sort of really ugly fight. I'm sorry. It's not going to happen The kind of person who acts like that (and has shown a pattern of same) is not going to stop acting like that just because you ask them nicely.
Perhaps you find that you really just kind of hate this woman, that the idea of being associated with her sickens you, and that you want to punish her for the way that she mistreated you in the past. This is reasonable. She has generated a toxic workplace environment, she's forced you to hide part of your life for fear of interacting with her, and now she's trying to make gains off of you. It's reasonable that you would feel this way. It's also highly likely to end badly if you actively pursue it. It's like the next option, but you have less moral high ground to work with, and people will be able to tell, to a degree.
Perhaps you believe that she is bad at her job, and you want to protect your workplace from this. In that case, there are other things that you can do. It is reasonable to believe this thing. Apparently, as a manager, she generated a sufficiently toxic workplace for you that you had to leave. "Not generating toxic workplace effects" is part of a manager's job. Understand, though, that this is going to be a matter of you trying to undermine the career of someone who is presumably higher-ranking, who you've been known to have trouble with in the past, and who is apparently reasonably good at playing political games. For a variety of reasons, this is not trivial.
So if you want to act, in a way that protects the company, the objective is to ensure that the people who matter know that she's slinging BS. First, don't make a big deal about it. This lady sounds like she thrives on drama. If the topic comes up in your presence, at a time when you might be reasonably expected to respond, then you can offer calm and quiet correction. Simply state that you have been attending, but that you do not believe that she is in your class. Keep it to simple facts that you know with absolute certainty to be true about your own experience, rather than speaking directly to her honesty or lack thereof. The calmer and less emotionally invested you can be about this, the better. Beyond that, you have a new manager. Speak to your new manager. Express concern. Your old manager is lying about this, and has shown what appears to be a pattern of lying about such things. This suggests that she may be misrepresenting both her own level of ability and the contributions she is providing to the company. That's a potential issue, especially if she gets herself into a position to make meaningful decisions for the company that she's not qualified to make. Acknowledge your own prior history with her, and the fact that it leaves you with some bias on the matter, and that as a result you're not necessarily the best person to push this up the chain further. Then leave it in their hands, and be prepared to back their play if they decide they're going to do something about it.
Possibly also throw in something about how you find it somewhat personally offensive, and it bothers you, but you don't know what to do about it. Then follow their advice.
That's how you handle issues with other managers who are not themselves reasonable people. You go to your manager first, you lay out the situation as best you can, and then you get ready to back their play, whatever it is.